Localization of eigenvector centrality in networks with a cut vertex

TL;DR

This paper investigates how eigenvector centrality localizes in networks with a cut vertex, revealing three types of localization and connecting eigenvector centrality to Katz centrality for better understanding.

Contribution

It introduces and characterizes two new types of localization phenomena in networks with cut vertices and relates eigenvector centrality to Katz centrality for improved interpretation.

Findings

Identifies three types of eigenvector localization phenomena.

Derives the relationship between eigenvector and Katz centrality.

Provides insights into the robustness of centrality measures in partitioned networks.

Abstract

We show that eigenvector centrality exhibits localization phenomena on networks that can be easily partitioned by the removal of a vertex cut set, the most extreme example being networks with a cut vertex. Three distinct types of localization are identified in these structures. One is related to the well-established hub node localization phenomenon and the other two are introduced and characterized here. We gain insights into these problems by deriving the relationship between eigenvector centrality and Katz centrality. This leads to an interpretation of the principal eigenvector as an approximation to more robust centrality measures which exist in the full span of an eigenbasis of the adjacency matrix.